Most web APIs are not designed with this use-case in mind. They're designed to facilitate web apps that are much more specific in what they're trying to present to the user. This is both deliberate and valuable; app creators need to be able to control the presentation to achieve their apps' goals.
REST API design is for use-cases where the users should have control over how they interact with the resources provided by the API. Some examples that should be using REST API design:
- Government portals for publicly accessible information, like legal codes, weather reports, or property records
- Government portals for filing forms and other interactions
- Open data initiatives like Wikipedia and OpenStreetmap
Considering these examples, it makes sense that policing of what "REST" means comes from the more academically-minded, while the detractors of the definition are typically app developers trying to create a very specific user experience. The solution is easy: just don't call it REST unless it actually is.